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Section 4.2 Row Operations as Matrix Multiplication (MX2)
Learning Outcomes
Subsection 4.2.1 Class Activities
Activity 4.2.1.
Let \(A=\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]\text{.}\) Find a \(3 \times 3\) matrix \(B\) such that \(BA=A\text{,}\) that is,
\begin{equation*}
\left[\begin{array}{ccc} \unknown & \unknown & \unknown \\
\unknown & \unknown & \unknown
\\ \unknown & \unknown & \unknown \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
\end{equation*}
Check your guess using technology.
Definition 4.2.2.
The identity matrix \(I_n\) (or just \(I\) when \(n\) is obvious from context) is the \(n \times n\) matrix
\begin{equation*}
I_n = \left[\begin{array}{cccc} 1 & 0 & \cdots & 0 \\ 0 & 1 & \ddots & \vdots \\
\vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right].
\end{equation*}
It has a \(1\) on each diagonal element and a \(0\) in every other position.
Fact 4.2.3.
For any square matrix \(A\text{,}\) \(IA=AI=A\text{:}\)
\begin{equation*}
\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
\end{equation*}
Activity 4.2.4.
Tweaking the identity matrix slightly allows us to write row operations in terms of matrix multiplication.
(a)
Create a matrix that doubles the third row of \(A\text{:}\)
\begin{equation*}
\left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 2 & 2 & -2 \end{array}\right]
\end{equation*}
(b)
Create a matrix that swaps the second and third rows of \(A\text{:}\)
\begin{equation*}
\left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 1 & 1 & -1 \\ 0 & 3 & 2 \end{array}\right]
\end{equation*}
(c)
Create a matrix that adds \(5\) times the third row of \(A\) to the first row:
\begin{equation*}
\left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right]
\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
=
\left[\begin{array}{ccc} 2+5(1) & 7+5(1) & -1+5(-1) \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]
\end{equation*}
Fact 4.2.5.
If \(R\) is the result of applying a row operation to \(I\text{,}\) then \(RA\) is the result of applying the same row operation to \(A\text{.}\)
Scaling a row: \(R=
\left[\begin{array}{ccc}
c & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)
Swapping rows: \(R=
\left[\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right]\)
Adding a row multiple to another row: \(R=
\left[\begin{array}{ccc}
1 & 0 & c \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)
Such matrices can be chained together to emulate multiple row operations. In particular,
\begin{equation*}
\RREF(A)=R_k\dots R_2R_1A
\end{equation*}
for some sequence of matrices \(R_1,R_2,\dots,R_k\text{.}\)
Activity 4.2.6.
Consider the two row operations \(R_2\leftrightarrow R_3\) and \(R_1+R_2\to R_1\) applied as follows to show \(A\sim B\text{:}\)
\begin{align*}
A
=
\left[\begin{array}{ccc}
-1&4&5\\
0&3&-1\\
1&2&3\\
\end{array}\right]
&\sim
\left[\begin{array}{ccc}
-1&4&5\\
1&2&3\\
0&3&-1\\
\end{array}\right]\\
&\sim
\left[\begin{array}{ccc}
-1+1&4+2&5+3\\
1&2&3\\
0&3&-1\\
\end{array}\right]
=
\left[\begin{array}{ccc}
0&6&8\\
1&2&3\\
0&3&-1\\
\end{array}\right]
=
B
\end{align*}
Express these row operations as matrix multiplication by expressing \(B\) as the product of two matrices and \(A\text{:}\)
\begin{equation*}
B =
\left[\begin{array}{ccc}
\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown
\end{array}\right]
\left[\begin{array}{ccc}
\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown
\end{array}\right]
A
\end{equation*}
Check your work using technology.
Subsection 4.2.2 Videos
Figure 43. Video: Row operations as matrix multiplication
Subsection 4.2.5 Sample Problem and Solution
Sample problem Example B.1.19.